Waves running up and down a hanging rope

Joe hangs a rope of length L = 100 m and mass m = 5 kg from a high beam. He adopts a coordinate system in which the height, y, is measured from the bottom of the rope. So y = 0 at the bottom and y = L at the top.

Joe gets in a crane, rides up to the middle of the rope (y = 50 m), and whacks the rope with a stick. He watches as waves run away from the middle: one runs up to the top, the other runs down to the bottom. Which wave reaches the end of the rope first?

     A.   Wave going up reaches top first

     B.   Wave going down reaches bottom first

     C.   Both waves reach the ends at the same time

1. Write an equation which yields the mass of the rope hanging below a point at height y.
2. Write an equation which yields the tension in the rope at a height y.
3. Write an equation which yields the speed of a wave in the rope at a height y.
4. Write an equation which yields the time it takes a wave at height y to travel through a teeny length of the rope, of size dy.
5. How long will it take the wave going up to reach the top of the rope?
6. How long will it take the wave going down to reach the bottom of the rope?

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.